Typhlops capitulatus

Number of Pages: 2Integrative BiologyGeological Science

Stability of a fermionic N+1N+1 particle system with point interactions

We prove that a system of $N$ fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical $m^*$. The value of $m^*$ is independent of $N$ and turns out to be less than $1$. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain.Comment: LaTeX, 29 pages, 2 figures; typos corrected, explanations and references added; to appear in Commun. Math. Phy

Stability of the 2+2 fermionic system with point interactions

We give a lower bound on the ground state energy of a system of two fermions of one species interacting with two fermions of another species via point interactions. We show that there is a critical mass ratio m_c \approx 0.58 such that the system is stable, i.e., the energy is bounded from below, for m \in [m_c, m_c^{-1}]. So far it was not known whether this 2+2 system exhibits a stable region at all or whether the formation of four-body bound states causes an unbounded spectrum for all mass ratios, similar to the Thomas effect. Our result gives further evidence for the stability of the more general N+M system.Comment: LaTeX, 12 pages; typos corrected, references and 2 figures added; to appear in Math. Phys. Anal. Geo

Typhlops schwartzi

Number of Pages: 2Integrative BiologyGeological Science

Triviality of a model of particles with point interactions in the thermodynamic limit

We consider a model of fermions interacting via point interactions, defined via a certain weighted Dirichlet form. While for two particles the interaction corresponds to infinite scattering length, the presence of further particles effectively decreases the interaction strength. We show that the model becomes trivial in the thermodynamic limit, in the sense that the free energy density at any given particle density and temperature agrees with the corresponding expression for non-interacting particles.Comment: LaTeX, 20 pages; final version, to appear in Lett. Math. Phy

R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators

It is shown that an elliptic scattering operator $A$ on a compact manifold with boundary with coefficients in the bounded operators of a bundle of Banach spaces of class (HT) and Pisier's property $(\alpha)$ has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of $A$ on the scattering cotangent bundle of the manifold avoids the right half-plane. This is deduced directly from a Seeley theorem, i.e. the resolvent is represented in terms of pseudodifferential operators with R-bounded symbols, thus showing by an iteration argument the R-boundedness of $\lambda(A-\lambda)^{-1}$ for $\Re(\lambda) \geq 0$. To this end, elements of a symbolic and operator calculus of pseudodifferential operators with R-bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on $R^d$ with operator valued coefficients.Comment: 21 page

Summing curious, slowly convergent, harmonic subseries

The harmonic series diverges. But if we delete from the harmonic series all terms whose denominators contain any string of digits such as "9", "42", or "314159", then the sum of the remaining terms converges. These series converge far too slowly to compute their sums directly. We describe an algorithm to compute these and related sums to high precision. For example, the sum of the series whose denominators contain no "314159" is approximately 2302582.33386. We explain why this sum is so close to 106 log 10 by developing asymptotic estimates for sums that omit strings of length n, as n approaches infinity. \ud \ud The first author is supported by a Rhodes Scholarship